Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the field of meromorphic functions on $F$ is isomorphic to the field of rational functions in one or two variables over $mathbb C$.
{"title":"On fields of meromorphic functions on neighborhoods of rational curves","authors":"Serge Lvovski","doi":"arxiv-2408.14061","DOIUrl":"https://doi.org/arxiv-2408.14061","url":null,"abstract":"Suppose that $F$ is a smooth and connected complex surface (not necessarily\u0000compact) containing a smooth rational curve with positive self-intersection. We\u0000prove that if there exists a non-constant meromorphic function on $F$, then the\u0000field of meromorphic functions on $F$ is isomorphic to the field of rational\u0000functions in one or two variables over $mathbb C$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the Bergman kernel. As an application, we characterize two dimensional ball quotients as the only Stein spaces with smooth finite type boundary and locally rational Bergman kernel. A key ingredient in the proof of the degree estimate is a new localization result for the Bergman kernel of a pseudoconvex, finite type domain in a complex manifold.
{"title":"Local algebraicity and localization of the Bergman kernel on Stein spaces with finite type boundaries","authors":"Peter Ebenfelt, Soumya Ganguly, Ming Xiao","doi":"arxiv-2408.13989","DOIUrl":"https://doi.org/arxiv-2408.13989","url":null,"abstract":"On a two dimensional Stein space with isolated, normal singularities, smooth\u0000finite type boundary, and locally algebraic Bergman kernel, we establish an\u0000estimate on the type of the boundary in terms of the local algebraic degree of\u0000the Bergman kernel. As an application, we characterize two dimensional ball\u0000quotients as the only Stein spaces with smooth finite type boundary and locally\u0000rational Bergman kernel. A key ingredient in the proof of the degree estimate\u0000is a new localization result for the Bergman kernel of a pseudoconvex, finite\u0000type domain in a complex manifold.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine a simple expression of the Picard-Fuchs system for a family of Kummer surfaces for all principally polarized Abelian surfaces. It is given by a system of linear partial differential equations in three variables of rank five. Our results are based on a Jacobian elliptic fibration on Kummer surfaces and a GKZ hypergeometric system suited to the elliptic fibration.
{"title":"Picard-Fuchs system for family of Kummer surfaces as subsystem of GKZ hypregeometric system","authors":"Atsuhira Nagano","doi":"arxiv-2408.14271","DOIUrl":"https://doi.org/arxiv-2408.14271","url":null,"abstract":"We determine a simple expression of the Picard-Fuchs system for a family of\u0000Kummer surfaces for all principally polarized Abelian surfaces. It is given by\u0000a system of linear partial differential equations in three variables of rank\u0000five. Our results are based on a Jacobian elliptic fibration on Kummer surfaces\u0000and a GKZ hypergeometric system suited to the elliptic fibration.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"400 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate various square functions on the unit complex ball. We prove the weighted inequalities of the Lusin area integral associated with Poisson integral in terms of $A_p$ weights for all $1
本文研究了单位复球上的各种平方函数。我们证明了在所有 1
{"title":"Weighted norm inequalities of various square functions and Volterra integral operators on the unit ball","authors":"Changbao Pang, Maofa Wang, Bang Xu, Hao Zhang","doi":"arxiv-2408.13726","DOIUrl":"https://doi.org/arxiv-2408.13726","url":null,"abstract":"In this paper, we investigate various square functions on the unit complex\u0000ball. We prove the weighted inequalities of the Lusin area integral associated\u0000with Poisson integral in terms of $A_p$ weights for all $1<p<infty$; this\u0000gives an affirmative answer to an open question raised by Segovia and Wheeden.\u0000To that end, we establish the weighted inequalities for Littlewood-Paley type\u0000square functions. As an interesting application, we obtain the weighted\u0000inequalities of the Lusin area integral associated with Bergman gradient. In\u0000addition, we get an equivalent characterization of weighted Hardy spaces by\u0000means of the Lusin area integral in the context of holomorphic functions. We\u0000also obtain the weighted inequalities for Volterra integral operators. The key\u0000ingredients of our proof involve complex analysis, Calder'on-Zygmund theory,\u0000the local mean oscillation technique and sparse domination.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, Miller Ross function with bicomplex arguments has been introduced. Various properties of this function including recurrence relations, integral representations and differential relations are established. Furthermore, the bicomplex holomorphicity and Taylor series representation of this function are discussed, along with the derivation of a differential equation. Finally, as applications some relations of fractional order derivatives and solutions for the bicomplex extension of the generalized fractional kinetic equation involving the bicomplex Miller Ross function are derived.
{"title":"Properties and applications of the Bicomplex Miller-Ross function","authors":"Snehasis Bera, Sourav Das, Abhijit Banerjee","doi":"arxiv-2408.13246","DOIUrl":"https://doi.org/arxiv-2408.13246","url":null,"abstract":"In this work, Miller Ross function with bicomplex arguments has been\u0000introduced. Various properties of this function including recurrence relations,\u0000integral representations and differential relations are established.\u0000Furthermore, the bicomplex holomorphicity and Taylor series representation of\u0000this function are discussed, along with the derivation of a differential\u0000equation. Finally, as applications some relations of fractional order\u0000derivatives and solutions for the bicomplex extension of the generalized\u0000fractional kinetic equation involving the bicomplex Miller Ross function are\u0000derived.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a new residue formula for integrals with singularities along affine hyperplanes. Our formula makes use of a notion for real matrices called stability which is inspired by ideas from total positivity.
{"title":"A residue formula for integrals with hyperplane singularities","authors":"Andrew O'Desky","doi":"arxiv-2408.12586","DOIUrl":"https://doi.org/arxiv-2408.12586","url":null,"abstract":"We prove a new residue formula for integrals with singularities along affine\u0000hyperplanes. Our formula makes use of a notion for real matrices called\u0000stability which is inspired by ideas from total positivity.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"16 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A novel asymptotic representation of the analytic solutions to a family of singularly perturbed $q-$difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated to the vanishing rate of the domains of the coefficients in the formal asymptotic expansion. On the way, a novel version of a multilevel sequential Ramis-Sibuya type theorem is achieved.
{"title":"On parametric $0$-Gevrey asymptotic expansions in two levels for some linear partial $q$-difference-differential equations","authors":"Alberto Lastra, Stephane Malek","doi":"arxiv-2408.12335","DOIUrl":"https://doi.org/arxiv-2408.12335","url":null,"abstract":"A novel asymptotic representation of the analytic solutions to a family of\u0000singularly perturbed $q-$difference-differential equations in the complex\u0000domain is obtained. Such asymptotic relation shows two different levels\u0000associated to the vanishing rate of the domains of the coefficients in the\u0000formal asymptotic expansion. On the way, a novel version of a multilevel\u0000sequential Ramis-Sibuya type theorem is achieved.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we consider some problems about a reflected graph map germ $f$ from $(mathbb{C}^n,0)$ to $(mathbb{C}^{n+1},0)$. A reflected graph map is a particular case of a reflection map, which is defined using an embedding of $mathbb{C}^n$ in $mathbb{C}^{p}$ and then applying the action of a reflection group $G$ on $mathbb{C}^{p}$. In this work, we present a description of the presentation matrix of $f_*{cal O}_n$ as an ${cal O}_{n+1}$-module via $f$ in terms of the action of the associated reflection group $G$. We also give a description for a defining equation of the image of $f$ in terms of the action of $G$. Finally, we present an upper (and also a lower) bound for the multiplicity of the image of $f$ and some applications.
{"title":"On reflection maps from n-space to n+1-space","authors":"Milena Barbosa Gama, Otoniel Nogueira da Silva","doi":"arxiv-2408.11669","DOIUrl":"https://doi.org/arxiv-2408.11669","url":null,"abstract":"In this work we consider some problems about a reflected graph map germ $f$\u0000from $(mathbb{C}^n,0)$ to $(mathbb{C}^{n+1},0)$. A reflected graph map is a\u0000particular case of a reflection map, which is defined using an embedding of\u0000$mathbb{C}^n$ in $mathbb{C}^{p}$ and then applying the action of a reflection\u0000group $G$ on $mathbb{C}^{p}$. In this work, we present a description of the\u0000presentation matrix of $f_*{cal O}_n$ as an ${cal O}_{n+1}$-module via $f$ in\u0000terms of the action of the associated reflection group $G$. We also give a\u0000description for a defining equation of the image of $f$ in terms of the action\u0000of $G$. Finally, we present an upper (and also a lower) bound for the\u0000multiplicity of the image of $f$ and some applications.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the relationship between the valuations of a germ of a singular foliation $mathcal{F}$ on the complex plane and those of a balanced equation of separatrices for $mathcal{F}$, extending a theorem by Mattei-Salem. Under certain conditions, we also derive inequalities involving the valuation, tangency excess, and degree of a holomorphic foliation $mathcal{F}$ on the complex projective plane.
{"title":"On a Mattei-Salem theorem","authors":"Arturo Fernández-Pérez, Nancy Saravia-Molina","doi":"arxiv-2408.10767","DOIUrl":"https://doi.org/arxiv-2408.10767","url":null,"abstract":"We investigate the relationship between the valuations of a germ of a\u0000singular foliation $mathcal{F}$ on the complex plane and those of a balanced\u0000equation of separatrices for $mathcal{F}$, extending a theorem by\u0000Mattei-Salem. Under certain conditions, we also derive inequalities involving\u0000the valuation, tangency excess, and degree of a holomorphic foliation\u0000$mathcal{F}$ on the complex projective plane.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study holomorphic actions of the complex multiplicative group on complex manifolds around a singular (fixed) point. We prove linearization results for the germ of action and also for the whole action under some conditions on the manifold. This can be seen as a follow-up to previous works of M. Suzuki and other authors.
本文研究复数乘法组在复数流形上围绕奇点（定点）的全态作用。我们提出了作用胚芽的线性化结果，以及流形上某些条件下整个作用的线性化结果。这可以看作是对铃木 M. 和其他作者之前研究成果的补充。
{"title":"On holomorphic $mathbb{C}^*$-actions","authors":"Víctor León, Bruno Scárdua","doi":"arxiv-2408.09625","DOIUrl":"https://doi.org/arxiv-2408.09625","url":null,"abstract":"In this paper we study holomorphic actions of the complex multiplicative\u0000group on complex manifolds around a singular (fixed) point. We prove\u0000linearization results for the germ of action and also for the whole action\u0000under some conditions on the manifold. This can be seen as a follow-up to\u0000previous works of M. Suzuki and other authors.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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